Talk:List of paradoxes

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This article is a fork of material originally contained in the article Paradox. --Jeffrey O. Gustafson - Shazaam! - <*> 03:05, 31 December 2005 (UTC)

Hey, please remove items in the article that do not belong - that are not paradoxes or paradoxical. The "Monty Hall Problem", for example, is not a paradox. It is not understood as one for those familiar with the problem, and it clearly does not fit the definition as stated in this Wikipedia article or elsewhere. In this case, even though many people "fall for it", it is a clear mathematical problem with a clear, logical, unambigious solution. Lets keep the list for paradoxes only! .......or, at least categorize these two: real or seemingly-real paradoxes VS. psuedo-paradoxes.

Absolutely correct. The Monty Hall problem is definitely not a paradox, and has been removed. MathStatWoman 03:57, 24 January 2006 (UTC)

We have some difficulties here. The twin paradox is not a paradox, since one twin experiences acceleration and the other does not. Will Rogers paradox is not a paradox. We need to clean up this article. MathStatWoman 08:57, 27 January 2006 (UTC)

Also: statistical "paradox": misinterpretation of correlation does not constitute a paradox! Please, let us clean up this page! I would start to do so, but I do not want to be unjustly accused of vandalism, so could we do it as a team? MathStatWoman 09:01, 27 January 2006 (UTC)

MathStatWoman seems to be using "paradox" to mean simply "contradiction". That's not the only meaning, and obviously not the one intended here. Monty Hall is definitely a paradox, whereas "defining sets of sets" does not lead to paradoxes (naive comprehension leads to a contradiction, and if you have an intuition that naive comprehension is true, then that fact will be paradoxical for you, but it doesn't have much to do with defining sets of sets). By the way, the term "real paradox" reminds me of the commercial that used to air on late-night TV in which "if you order now, we'll send you these genuine faux pearls!". --Trovatore 16:41, 27 January 2006 (UTC)

Yes, I think people are being absurd here. Almost none of the mathematical paradoxes are actually paradoxes in the strictest sense: Tarski-Banach, the Monty Hall problem or the Birthday Party paradox. However, they are called paradoxes because strange outcomes occur from the given situation. This is a perfectly normal use of the word paradox in mathematics. I don't think anything would be left in the Maths/Stats part (except maybe some Bayesian/frequentist wrangling about the two envelopes paradox) if we removed things on these grounds. --Richard Clegg 14:43, 30 January 2006 (UTC)

Actually I like my genuine faux diamonds. Don't you love these debates based on semantics? What fun. MathStatWoman 00:20, 28 January 2006 (UTC)

Maybe I'm missing something, but this list of paradoxes doesn't appear to contain the Prisoner's Dilemma. Surely PD is a bona-fide paradox, much more so than the examples correctly identified above as merely mundane faulty reasoning. A genuine paradox is non-resolvable, and the one-round PD qualifies in this respect, along with the similar 'Trajedy of the Commons' formulation. MelbournePaul 22:00, 30 January 2006

How is Unintended consequence a paradox??? 59.167.131.98 12:50, 3 June 2006 (UTC)

I had removed the "missing square puzzle", which is an optical illusion, but some genius has put it back in, giving the reason, and I quote Many of the listed things here are not really paradoxes. Help!!! --Arno Matthias 11:09, 1 October 2006 (UTC)

That was me. Taking just some examples, the missing dollar paradox, Hodgekinson's paradox, Monty Hall problem, Twins, Smale, Low Birth Weight Problem, everything (I think) in the "Geometry and Topology" are not really paradoxes. I would not be surprised if the majority of the things on this page are not really paradoxes. The point is that they appear paradoxial. --Richard Clegg 12:25, 1 October 2006 (UTC)

Oh for pete's sake - if you already know it doesn't belong here, then why do you put it in again?? Is this your idea of cleaning up this mess? The article is called "List of paradoxes", and not, for example, "List of puzzles that some people find difficult to solve" or "List of optical illusions". If some people want to keep these items in they should at least be in a different section "Things that may look like paradoxes but really aren't" - agreed? --Arno Matthias 11:38, 2 October 2006 (UTC)

Arno... please try and stay polite. Look at the rest of the discussion on this talk page. How is the "missing square puzzle" different to the "missing dollar paradox" for example. If we followed your idea I think most of the paradoxes on this page would not be here. Most of the things on this page are not paradoxes in the strictest sense. --Richard Clegg 11:42, 2 October 2006 (UTC)

Contents 1 Categorization 2 Petronius' paradox 3 Clasifying paradoxes 4 benford's law 5 Possession paradox

[edit] Categorization

The categorization of paradoxes here is pretty rough and could use some work. Let's remember that paradoxes are only paradoxes-in-a-theory, so we have semantic paradoxes that are paradoxes due to our theory of truth, and set-theoretic paradoxes that are subsumed within our set theory. Things like the Monty Hall problem are properly called paradoxes as long as we recognize that they are paradoxes within decision theory. KSchutte 19:54, 12 March 2006 (UTC)

[edit] Petronius' paradox

I did a little searching around, and found that this is not his paradox. Unless someone can prove me wrong it should be removed.

[edit] Clasifying paradoxes

There's a continual problem with this page that some of the things listed are "paradoxes" that is they have no resolvalbe answer e.g. Russell's paradox and its answers. Some seem paradoxes because the explanation has a "trick" Horse paradox. Some are not paradoxes but seemingly unlikely outcomes of a theory Twin Paradox or Banach-Tarski Paradox. This page occasionally suffers deletions because someone says "that's not a paradox" of something in the last two classes (usually the easier to understand ones). Would it be helpful to try to classify paradoxes as to their status? It would lead to countless disagreements I have no doubt. --Richard Clegg 12:53, 1 October 2006 (UTC)

I see no such distinction. A paradox is, by definition, an apparent contradiction that breaks a natural-seeming intuition. Russell's paradox is precisely of this sort, as is Banach-Tarski. --Trovatore 02:44, 2 October 2006 (UTC)

That is not how some people would see a "true" paradox. There is a fundamental difference between the two. Banach-Tarski is just a rather strange consequence which you might not expect -- it is peculiar but not paradoxical in a true sense. Russell's paradox leads to an undecidable consequence and shows an underlying problem with the system. It is more than a "strange consequence" it's a flat out inconsistency. Russell's paradox led to a reformulation of set theory because it highlighted a flaw with the theory of the time -- the axioms of the system were not consistent. Banach-Tarski did not lead to such a reformulation of set theory it was just one of the many strange consequences of the Axiom of Choice. Perhaps an even clearer example is Simpson's Paradox which is actually clear to anyone with basic maths once it's explained. --Richard Clegg 07:24, 2 October 2006 (UTC)

Sure. See Quine's classification, for example. But I underline that they are all still called paradoxes, not apparent paradoxes, so I don't exactly agree with the sentence you are trying to add. 192.75.48.150 13:59, 2 October 2006 (UTC)

OK -- Quine's classification is a useful one. I like it. What I'm trying to avoid is the continual problem of people who try to remove things off this page because "they're not paradoxes". I have added a link to Quine's classification above. I agree they're all still called paradoxes but still we get people deleting things because "they're not paradoxes" -- this particularly happens with the easier to understand ones. --Richard Clegg 14:45, 2 October 2006 (UTC)

One could argue here that there still is no fundamental difference, though. The "underlying problem with the system" is as much a consequence of the assumption that this kind of thing cannot happen, as it is actually happening. Indeed, all the self-referential paradoxes (which include all the Liar paradoxes) illuminate how the intuition about reference, use/mention, etc., is not always reliable; "a rather strange consequence which you might not expect." I point out the analogy of those who first dabled in non-Euclidean geometry and thought they disproved it because they (thought they) contradicted something Euclidean. Of course they didn't since the axioms themselves were inconsistent, but their intuition was assaulted.

Exapmple: note also that a contradiction is not found in the sentence "This sentence is false", because it is not both true and false. Indeed, it is neither. It is counterintuitive, though, as we tend to think any well formed sentence has to be either true, or false. The contradiction comes only when we assume this has to be the case. But as you allude to, it doesn't.

About the use of "clear": We need to be careful to distinguish between "logically rigorous" (which your usage would support), and "intuitively obvious" (which it would not). Only careful familiarity with the rigor of the explanation can update one's intuition; that is a difficult process. This gets at use/mention again: I could accurately reword the end of your Simpson's sentence as follows: "Perhaps an even clearer example is Simpson's Paradox, the truth of whose unintuitive result is actually clear [...], once it's explained" The result can still be unintuitive, even if it is accepted. So I disagree as well with the sentence. Baccyak4H 14:34, 2 October 2006 (UTC)

Oh god, you're a philosopher aren't you.  :-) What I'm trying to get at is that there is a fundamental difference between, say the "Horses are all the same colour" paradox (a false proof that all horses are the same colour as each other) this is just faulty reasoning, someone explains it and you go "oh yes, I see", the "Simpson's paradox" which is an unlikely outcome which seems counter-intuitive and the "Russell paradox" which indicated a fundamental flaw in the underlying axiomatic system. There are, of course, intermediates -- the Schrodingers cat paradox may be a counter-intuitive result of QM or a fundamental flaw in QM (maybe even some people would say it is just faulty reasoning) depending on just who you ask. --Richard Clegg 14:45, 2 October 2006 (UTC)

Well, I would argue that it is not fundamental, but a matter of degree. To the horse colors (low degree), a response might be "no, horses can be different colors, yuor argument is fallaceous." To something else, there might be a lot of handwaving and headscratching. "How can that be????"

But to be fair, it is not clear to me where the dividing line between paradox and outright fallacy is. People's intuitions are honed differently, etc. So there is some ambiguity here.

As for the wording, I would argue that all paradoxes have in principle a clear resolution, but I use clear to mean rigorous. But some are much harder to accept on a gut level (as opposed to a reason level). So let's go slowly here; your clarification is already a significant improvement.

I am not a professional philosopher, but I do "love knowledge". And where I come from, that is a compliment.  :-) Baccyak4H 15:12, 2 October 2006 (UTC)

The dividing line, for me as a working mathematician, is whether or not the paradox indicates that your system is itself flawed. The Horse Colour paradox is fun because it's a nice example of trying to find a problem with the reasoning. It's like all of those proofs that 1 = 2 which rely on a "trick". It's an intellectual game of "spot the deliberate mistake". The Russell Paradox indicated a problem with the formulation of set theory and hence the underlying mathematics had to be developed to avoid the paradox. In other words, the clear resolution to the Russell Paradox was to develop a new form of mathematics where the Russell Paradox wasn't possible. The clear resolution to the Horse paradox was to say "you made a mistake here at step 2". This to me, is a fundamental difference. --Richard Clegg 15:46, 2 October 2006 (UTC)

I think we have to get clear about which "system" you're talking about, in the case of the Russell Paradox. The system directly refuted by Russell was Fregean set theory. Fregean set theory was flat-out falsified by the Russell Paradox, crushed beyond hope of resurrection. But Fregean set theory had never been "the" system; mathematics had never been based upon it, so a new form of mathematics was not needed, just a new understanding of sets (possibly not that new after all; it's a point of debate whether Cantor's unformalized conception was closer to Frege's or to the modern one).

But we don't call RP a "paradox" because it refuted Fregean set theory. Fregean set theory was simply in error; no paradox there. It's a "paradox" because it violates a natural-seeming intuition. In that sense it's just like Banach-Tarski. --Trovatore 16:02, 2 October 2006 (UTC)

Maybe my undersanding of history is in error here. As I understood it, Fregean set theory was the set theory of the day -- it was what people used to work with sets apart from a few people -- sure mathematics itself wasn't based on it but it was a respected branch of mathematics. Russell's paradox crushed it as you say, leading to new formulations of set theory. We call RP a paradox not just because it violates an intuition but because it shows up a flaw in a theory. There is no Russell Paradox in ZF set theory for example. This is different to the B-T paradox which, while intended to prove the incorrectness of the Axiom of Choice (as I understand it) in fact was just accepted as "one of those weird things". B-T paradox can be accepted within those formulations of set theory where it occurs, without destroying the theory. No theory which admits a Russell type paradox can be admitted within mathematics unless one is prepared to take an extremely anarchic definition of what is mathematics. --Richard Clegg 16:15, 2 October 2006 (UTC)

I'm no historian, but I kind of do think your history is in error. It's sort of a common error, though.

What is true, I think, is that mathematicians of the time used a sort of intuitive notion of set that can be roughly elucidated as follows:

The extensional notion of a set (determined by its members), and the intensional notion of a class (determined by its definition), are not really different, and may be uncritically interchanged.

This intuition, as we now know, was simply wrong. While there had been earlier indications of problems with it, it was RP that finished it off. Now, there was another intuition that had perhaps never been formalized, that went something like this:

Sets of points in R³ are like physical objects, to the extent that it should be possible to assign to every such set a "volume" or "mass" that behaves the way we expect it to, based on our physical experience

This latter intuition was also wrong, and we know it because of BT. These cases, to me, seem very closely parallel. --Trovatore 17:47, 2 October 2006 (UTC)

I think we will have to agree to disagree on this one -- the cantor middle third set would disprove that intuition about sets well before BT. However, both cases I hope you would agree, are very very different to the "All horses are the same colour" type paradox which is simply a "quickness of the hand deceives the eye" type "fake" proof. --Richard Clegg 17:56, 2 October 2006 (UTC)

Yeah, I guess I agree that the "horse" and "2=1" proofs are a different sort of thing from (not "different to", please! after all, I avoid saying "different than") the BTP and RP. However I don't really buy the thing about the middle thirds set--that just has zero measure, and there's nothing particularly unintuitive about that, as far as I can see. Not being able to assign a physics-respecting measure, even to elements of a partition into finitely many pieces, is much more troublesome. --Trovatore 05:27, 3 October 2006 (UTC)

[edit] benford's law

I don't know much about paradoxes and was just browsing this page, but it seems to me that Benford's Law should not be on this list. I don't think it is a paradox, but then again I'm no logician. I think if something is a scientific "law" (as the name purports), then by definition it cannot be a paradox. Thoughts?

See the above discussions and the wording at the top of the page about many of the paradoxes having a clear resolution. --Richard Clegg 11:03, 14 October 2006 (UTC)

[edit] Possession paradox

Has this paradox been included on the list - "a claim to having a right is only required when the subject does not have the object of that right". - Shiftchange 03:32, 3 November 2006 (UTC)