Welcome to the mathematics section
of the Wikipedia reference desk.

Select a section:

WP:RD/MA

Want a faster answer?

Main page: Help searching Wikipedia

How can I get my question answered?

Select the section of the desk that best fits the general topic of your question (see the navigation column to the right).
Post your question to only one section, providing a short header that gives the topic of your question.
Type '~~~~' (that is, four tilde characters) at the end – this signs and dates your contribution so we know who wrote what and when.
Don't post personal contact information – it will be removed. Any answers will be provided here.
Please be as specific as possible, and include all relevant context – the usefulness of answers may depend on the context.
Note:
- We don't answer (and may remove) questions that require medical diagnosis or legal advice.
- We don't answer requests for opinions, predictions or debate.
- We don't do your homework for you, though we'll help you past the stuck point.
- We don't conduct original research or provide a free source of ideas, but we'll help you find information you need.

Ready? Ask a new question!

How do I answer a question?

Main page: Wikipedia:Reference desk/Guidelines

The best answers address the question directly, and back up facts with wikilinks and links to sources. Do not edit others' comments and do not give any medical or legal advice.

February 3

Bhargava-Hanke 290 theorem

The 290 theorem is cited in that article as "Invent. mth., to appear" and cited in various places[1] as "Invent. Math. 2005". I was hoping to update the article with an exact reference but don't seem able to find the paper through Inventiones' own search form.[2] (other searches don't work either). Does anyone know how to find if the paper was actually published? There's no doubt of its validity, it is just a question of a proper cite, though maybe I shouldn't care since it's a Springer journal. Thanks. 73.93.153.74 (talk) 01:35, 3 February 2020 (UTC)[reply]

Apparently it has not yet been published. On his own website, Jon Hanke writes: "the paper should appear in the near future";[3] the statement has not been updated since September 21st, 2011. His "Math Papers" page only mentions the preprint. --Lambiam 08:54, 3 February 2020 (UTC)[reply]

The proof of the theorem was announced in Fall 2008, so 2005 is made of whole cloth. --Lambiam 09:06, 3 February 2020 (UTC)[reply]

I found a paper from last year that was still citing the preprint, so presumably if it has been published then it was 2019 at the earliest. It's an awkward situation as far as WP:RS is concerned since WP articles are only supposed to cite articles published in peer reviewed journals and not random web pages. (An over simplification but that's the gist of it in this case.) On the other hand it does seem like a significant result and a number of papers which have appeared in peer reviewed journals do cite it. But the article only says the proof was announced, so maybe just cite one of the published papers just mentioned to support that weaker claim. One of the objectives of WP:RS is to keep Joseph T. Crackpot from adding to the article on the Collatz conjecture that he's proved it and citing his post on blogspot.com as his source. So while there may be no doubt of the validity of the Bhargava-Hanke result, it sets a bad precedent when a self-published source is used as the only reference. Btw, the article has an ESL vibe to it imo; anyone feel like doing a copy edit on it? --RDBury (talk) 13:49, 3 February 2020 (UTC)[reply]

Apparently, Bhargava used the occasion of his receiving the 2005 SASTRA Ramanujan Prize to announce that he and Hanke had cracked the 290 conjecture; the preprint with the details of the proof followed later.[Article (paywall)]&[PDF] This can be used to source the first sentence of the last paragraph of the lead. If we replace the sentence about the proof going to appear in Inventiones Mathematicae by "A write-up of the proof is available as a preprint." and strike "Invent. Math., to appear." from the ref, the primary source can serve as acceptable direct evidence for this statement (that is, not requiring analysis or interpretation). --Lambiam 15:30, 3 February 2020 (UTC)[reply]

Thanks all. I wouldn't worry about the RS angle for this. The 290 theorem was part of Bhargava's Fields medal citation and a zillion other sources cite it too, so I'm sure it's fine. Perelman's proof of the Poincaré conjecture was never in a journal either, but his being awarded the Fields medal and the Millenium prize for it is good enough documentation that he really did it, even though he declined both awards. In the case of the 290 theorem, there was just what looked like an outdated reference to a paper then still in Inventiones' pipeline, so I thought it would have appeared by now. Apparently not, but no big deal under the circumstances. Added: for most math topics I wouldn't worry about formalities too much. Joseph T. Crackpot's work is focused mostly on a few major open problems like the Collatz conjecture. Topics with less glamour don't attract as many unreliable claims, so we can rely more on editorial judgment to identify the stuff that readers should want to see. 2601:648:8202:96B0:0:0:0:E118 (talk) 19:56, 3 February 2020 (UTC)[reply]

I agree that once you get away from the top five most famous unsolved problems you don't get many outlandish claims. But we have to deal with Mr. Crackpot at the help desk occasionally and my experience is that it's not wise to underestimate his persistence. The idea of relying on the RS policy rather than editorial judgment is WP editors can just point to the policy when they remove questionable claims; this helps avoid arguments. My worry was that Mr. Crackpot would point to the self-published cite in this article and say "This reference was self-published so why can't I use my self-published paper?" In any case, I'm going to go ahead and add Lambiam's information and cite into the article, and that should resolve everything. --RDBury (talk) 03:22, 4 February 2020 (UTC)[reply]

Your edit is fine, and those situations with J. T. Crackpot do come up once in a while, but not often enough that we should make unaffected articles suffer for them. In the case of the 290 theorem, you could also mention that the IMU noted it when they gave Bhargava the Fields medal.[4] That establishes its significance, among other things. And I'm fine with citing Crackpot's self-published preprints in Wikipedia, conditional on Crackpot getting the Fields medal or Millenium prize the way Bhargava and Perelman did for their preprints, lol.

More seriously, I'm satisfied if some established experts take a claimed result seriously, like here, citing blog writeups by some noted specialists. Wikipedia isn't a formal system--it's a textual work written for human readers and its value is in the relevant knowledge that it delivers to those readers. Its policy machinery is a means to that end, not something to pursue for its own sake. It's consistent with other observed practice to be stricter about the Collatz conjecture than about less flaky topics. See e.g. here, scroll down to "Editorial policy on submissions concerning famous problems". Just go by common sense and experience. 2601:648:8202:96B0:0:0:0:E118 (talk) 07:10, 4 February 2020 (UTC)[reply]

I was never saying that the link should be removed, but that it should be supported by reliable sources that support its significance as you said, so I think we're pretty much in agreement. And I agree that policy machinery us just a means to an end and not to be pursued as an end in itself. But I've experienced first hand how useful that machinery is and how important it can be to resolve arguments. I've been accused of holding a personal grudge against another editor for removing his amateurish original proof of some theorem (in geometry, I think). I've weighed in to delete in a WP:AfD discussion about an article that someone added about his deceased father whose claim to fame was he was the police chief of a small town. In such cases the best way of proceeding is to say "Well, Wikipedia has a strict policy that you need to cite reliable sources, otherwise the material can be deleted." Otherwise you have to say "Well, I've looked at your proof and it seemed clumsy and possibly erroneous, so I didn't think it should be in the article," or "Well, I know you want to honor your dad, but I don't think he was notable enough merit his own Wikipedia article." The reply would inevitably be "That's just your opinion, so I'm adding it back in." I was also active on Usenet when that was a thing, where they didn't have such policies, and saw how it attracted cranks and how prolific they can be; see Usenet personality. Wikipedia is different from a journal and its editorial policy has to differ accordingly. First, Wikipedia has an egalitarian structure where (theoretically) a high school student's contributions are given the same weight as John Conway's, while journals have a more hierarchical structure. Also, journals pay people to evaluate primary sources, but Wikipedia is essentially an all volunteer effort and the assumption is that volunteers will have neither the time, inclination, nor expertise for such tasks. This is why WP's policies demand secondary or tertiary over primary sources. So yes, policies are a means to an end, but if the end is to create a public source of reliable information, free of crank theories, fake news and other cruft, all based on an "everyone has equal say" philosophy, then yes you do need policies that are strictly adhered to. No policy can cover every eventuality and this case does seem to fall between the cracks, which is why some judgement needs to be applied in this case, but in such cases the policies should still be followed as much as reasonably possible. Just going by common sense and experience works when you have fact checkers and referees to fall back on and an editorial board to settle disputes, but it doesn't really work for Wikipedia, though it may seem like it should. I'm not claiming that Wikipedia's system or policies are perfect, far from it and that's one reason I'm no longer an active editor, but it has been extraordinarily successful as it is and it has escaped much of the recent criticism that has been directed at other platforms (Facebook, Twitter, etc.) about promoting conspiracy theories and extremist movements. --RDBury (talk) 19:44, 4 February 2020 (UTC)[reply]

February 4

Likeliest number of chess sets

The Lewis chessmen hoard contains 60 major chess pieces: 8 kings, 8 queens, 16 bishops, 15 knights and 13 rooks (which are called warders). They are listed here. They show no signs of colour. There are also 19 pawns so ignore them. If they came from four chess sets (as is often supposed) there would be 1 knight and 3 rooks missing but the individual pieces vary a lot from each other and so any definite allocation to sets is not possible. Some experts think that, on stylistic grounds, they may have come from five sets (or even more, but a small number anyway). It seems to me intuitively that it would be very unlikely to get such a distribution of pieces if they had come at random from five or more sets. Has this been looked at statistically? Thincat (talk) 09:53, 4 February 2020 (UTC)[reply]

I'm not aware of any such study and it would surprise me if there was one. I don't see any reason the pieces would have been taken at random (i.e. as independent draws in an urn model with urns consisting of chess sets). They'd more likely be correlated. 2601:648:8202:96B0:0:0:0:E118 (talk) 10:24, 4 February 2020 (UTC)[reply]

I think that for statistical modelling purposes it is a reasonable assumption that the pieces that went missing were like random draws from an urn with pieces, originally consisting of n complete sets of 16 officers. --Lambiam 14:13, 4 February 2020 (UTC)[reply]

Under that assumption, I compute the probability that four complete sets of officers, after losing 4 pieces, end up with the composition of the hoard as being equal to exactly 0.01410188612726952. For five complete sets of officers from which 20 pieces were lost, I find approximately 0.0033354 (or, exactly, 147396219858000/44191451777652179). The latter is roughly one quarter as likely, not dramatic enough a reduction to override considerations of style. For six sets I get about 0.0018358. It is furthermore conceivable that at some time complete sets were used for playing, which however were already heterogeneous because missing pieces had been replaced by pillaging artistically less accomplished and possibly also incomplete seta. --Lambiam 16:25, 4 February 2020 (UTC)[reply]

Thank you for those calculations. As you say, it doesn't make five sets look very much less plausible than four. Pondering on your reply, I think I now have a clearer view of what I should have asked this morning. Consider five complete sets of major pieces – 80 pieces – where all the pieces of each type are identical (and the same colour). We draw out 60 pieces. What is the chance we have 8 or less kings and queens and 16 or less of each of the other types of piece? Now, very much an aside – For nearly 200 years only 59 pieces were known until last year another piece, probably from the hoard, turned up. If it had been a king, queen or bishop the four set theory would have been disproved. (Or, more likely, people would have said the new piece wasn't from the hoard at all.) In fact it was a warder (rook) so everyone was happy, especially the person who was then able to sell it for £735,000! Thincat (talk) 19:43, 4 February 2020 (UTC)[reply]

The new condition includes all previously included compositions and many, many more, so it should not be surprising that it comes out much higher. I get about 0.22020. --Lambiam 20:36, 4 February 2020 (UTC)[reply]

Thank you again. Going back to my elementary statistics, I think that suggests that, given a null hypothesis of five sets, I should not be rejecting that on grounds of probability. I accept, of course, that the loss of pieces may have been selective and not random. Thincat (talk) 09:11, 5 February 2020 (UTC)[reply]

The situation is similar to the German tank problem (not sure why it was not linked yet): doing the math is the easy part, but choosing the initial hypotheses is hard/controversial because Bayes' theorem requires a prior probability.

For instance, reading the article's account of the discovery of the pieces, I would wildly speculate that the sets were complete or almost complete (trader's stock, never used) and buried in the sand dune, the 1831 guy dug up all he could but missed a few, someone else re-dug the same dune and found the rest, of which one recently resurfaced. If that story is true (that's a big if) it would make sense that only few pieces were missed in the first dig, so that weighs in favor of the four-set theory (missing 4 pieces) instead of the 5-set theory (missing 20 pieces). In the Bayesian framework, that would mean lending more initial credence to the four-set theory than to the five-set theory before doing the calculations above (which amplify the effect).

My narrative is a bit silly but the point is you have to consider what narrative your statistics are based on. Tigraan^{Click here to contact me} 16:51, 5 February 2020 (UTC)[reply]

February 5

Godel's Proofs: Implications for Mathematics

I'm reading a very interesting book: "The Information: A History, A Theory, A Flood" by James Gleick for an anthropology seminar. It is an historical overview of the concept of information from the Greek philosophers to the Internet. I like it a lot. However, when the author discusses Godel he makes what I think are some fundamental errors. However, I'm not a mathematician and while I have a good working knowledge of logic and set theory Godel's proofs have always been difficult and I want to reach out to those of you who are mathematicians to make sure I'm not missing something before I bring this up in the seminar.

My understanding is the first proof proves that any system that can express basic (e.g., Peano) arithmetic will have some theorems that are valid but not provable and the second proof is that one can't prove a system to be consistent within that system, one needs a meta-system. However, what Gleick says seems significantly different.

When he is talking about a scientist who encountered Godel's proofs as a young man he says (bold is mine):

"This [Godel] was heady stuff for the boy, who followed the authors through their simplified but rigorous exposition of Gödel’s “astounding and melancholy” demonstration that formal mathematics can never be free of self-contradiction. The vast bulk of mathematics as practiced at this time cared not at all for Gödel’s proof. Startling though incompleteness surely was, it seemed incidental somehow—contributing nothing to the useful work of mathematicians, who went on making discoveries".

He's not stating what the person thinks because there are other parts of the book where he talks about Godel and talks about how he showed that math was "inconsistent". My understanding is that there is a major difference between saying a system is incomplete (a consequence of Godel) vs. saying it is inconsistent (not free of self contradiction). Once you have an inconsistency your system is worthless because you can prove anything. (I also don't agree with his assessment about Godel's impact on mathematics in general but that's more of a subjective issue). Then later in the book he says when talking about a scientist who came across Godel's proofs that the scientist wondered:

"Was Gödel incompleteness related to Heisenberg uncertainty?"

which made me shout as I was reading: NO! In Linear Algebra and Its Applications by Gilbert Strang on p. 250 at the end of a chapter introducing Eigenvectors Strang has a little aside about how "the uncertainty principle follows directly from the Schwartz Inequality". Nothing at all to do with Godel. Does what he said in either case make some sense that I'm missing? --MadScientistX11 (talk) 03:37, 5 February 2020 (UTC)[reply]

First, a spelling gripe: Either Gödel with the umlaut or Goedel with an e, but never Godel.

Second, you're quite right. Gödel did not show (and certainly did not believe) that mathematics is inconsistent. What he did show is that (subject to certain conditions), given a formal theory T, if T proves that T is consistent, then T is in fact inconsistent. There's a lot to unwrap there, as the kids say, so it's not too surprising that people elide some of it and just jump to the conclusion that everything is inconsistent, whatever that's even supposed to mean.

Third, I certainly don't see any connection with Heisenberg uncertainty. But the two results feel similar to a lot of people, being "limitative" results of a sort. --Trovatore (talk) 03:45, 5 February 2020 (UTC)[reply]

The bit you've "bolded" is, I think, a misunderstanding. Godel didn't show that mathematics was inconsistent. Rather, he showed that if you have a logical system that is powerful enough to do ordinary arithmetic whilst still being consistent, there are valid statements that can be expressed in the system that can't be proven within the system.

On the physics, I'm not able to comment with confidence. RomanSpa (talk) 15:29, 5 February 2020 (UTC)[reply]

I should also say that Godel's results in this area don't strike me as being particularly "limitative". It helps that I'm a neo-formalist. RomanSpa (talk) 15:35, 5 February 2020 (UTC)[reply]

Thanks to everyone for those great answers. On the name: yes I knew I needed an umlaut but didn't know how to make one. I agree with RomanSpa that the theorems aren't that limiting. That's another small gripe I have with the book: he describes the work of Gödel, Turing, and Church and sort of leaves it at that as if they discovered these problems which math never resolved but then people just went on using math anyway when in reality (at least based on my limited understanding) all their work ultimately led to ZFC set theory which is IMO one of the most amazing accomplishments in math or science, it provides an incredibly rigorous foundation to build the rest of standard mathematics on. It probably sounds like a terrible book but actually it's quite good, he just misunderstands some of the issues with math and logic something I think is unfortunately common in the humanities. I think in the Stanford Encyclopedia of Philosophy they have a whole section of either Gödel's or Turing's entry devoted to the ways people have misinterpreted the results. Anyway, thanks again for the excellent and quick responses and for giving me an "ö" that I can copy paste from now on when I need it ;-) --MadScientistX11 (talk) 16:49, 5 February 2020 (UTC)[reply]

It's interesting that the book you're reading is interesting, because I read Gleick's biography of Isaac Newton a couple of years ago and it struck me as quite unmemorable. The writing was pedestrian; I'm surprised I even finished it. -- Jack of Oz ^{[pleasantries]} 16:51, 5 February 2020 (UTC) [reply]

Just for future reference, for me the easiest way to get spell difficult names like Gödel, Erdős, Plücker, Ørsted etc. is to type your best approximation in to WP's search box, let the autocorrect/redirect system do its magic to find the article on the person, then just copy and paste. It even works for non-mathies like Erdoğan; the media constantly get either the spelling or pronunciation wrong. --RDBury (talk) 20:04, 5 February 2020 (UTC) [reply]

Gleick has written good stuff about physics, and mathematical logic is an area where physicists tend to get things wrong ;-). Our article Gödel's incompleteness theorems explains the theorems in more detail, and Torkel Franzén's book Gödel's Theorem: An Incomplete Guide to its Use and Abuse has a lot of examples of mistaken things that have been published about the topic. Here (by George Boolos) is an explanation of the Second Incompleteness Theorem in words of one syllable. Fwiw there is no significant connection between Heisenberg uncertainty and mathematical incompleteness. 2601:648:8202:96B0:0:0:0:E118 (talk) 21:25, 5 February 2020 (UTC)[reply]