Talk:Hilbert's problems

message: I see in the archived talk that there was a long discussion about whether problem 6 was mathematical. Now it's 'Unresolved' in the table but it is not clear if there is a difference between this status and the 'Open' of several other problems. --Angelastic 06:38, 1 May 2007 (UTC)
replies: text: Any objections to this change?  If not I will make it reasonably soon.  --Vaughan Pratt (talk) 11:02, 7 January 2009 (UTC)
replies: text: What printed sources are there to which we might turn to help editors make a decision? Bill Wvbailey (talk) 14:51, 7 January 2009 (UTC)

text: My comment is years later than those above, but I believe the writers make a fundamental (and pretty obvious) blunder in assuming that Hilbert's use of the term 'physics' was (or should be) open-ended. What was he talking about? Either he was talking about the physics known to him OR he was talking about some nebulous universe of possible extensions of the understood physics of the day, with new (unknown with all sorts of possible structure) laws, relationships and structure. I submit that the latter possibility is very unlikely (but if it is, then clearly the conjecture is too vague to be resolved). I suggest that all uses of the word "physics" be replaced by the term "[Classical] physics" since its almost certain that is what he meant. Actually, I doubt the square editorial brackets are necessary, but for quotes should be inserted. Our purpose here is, after all, to convey meaning, not random sequences of characters on a page. Given that he meant classical physics, the 6th problem HAS been solved - it is neither open nor unresolved.173.189.79.42 (talk) 23:29, 15 May 2015 (UTC)
thread_title: Unresolved or Open?

message: The Mathematical Gazette, March 2000 (page 2-8) "100 Years On" seems to be a dead link Randomblue 07:43, 30 June 2007 (UTC)
replies: text: I've removed it. --Zundark 07:53, 30 June 2007 (UTC)
thread_title:  Dead link 

message: Is no one else bothered by the language and tone used in this article, which is 
obfuscatory, stilted and jargony?  Some examples:
"This might be put down to the eminence of the problems' author."  
Why not just "due to Hilbert's eminence"?
"would go on to lead" Why all these conditionals?  Hilbert DID go on to lead etc. 
"On closer examination, matters are not so simple."
What??  Whose examination? What matters?  Is the author trying to say that Hilbert's reputation as a mathematician is not the only reason the Hilbert problems are influential?  Yes; probably the fact that most of the problems were deep and interesting was worth as much or more than Hilbert's reputation in the mathematical community.  But I have read the paragraph several times and am still not sure what role this sentence is supposed to play.  
"The mathematics of the time was still discursive."
Again, what?? I had no idea what this sentence was supposed to mean until I clicked on
the footnote.  Putting in footnotes to clarify the meaning of your sentence is classic 
bad writing.  All of the footnotes should be assessed and either reincorporated into 
the text or removed.
:Well, I'm sorry you don't like the writing, which is mine. By the way, you have just used a classic "passive construction". I looked at this last night and was surprised at how little has changed in this article, recently. By simply listing lots of objections, you are registering disapproval. That's about all. The mathematics of 1900 was "discursive". That's the precise word. I can amplify, for example with the way Poincaré wrote. Charles Matthews 08:46, 30 September 2007 (UTC)
replies: text: Oh well, if you know the Constance Reid biography, you'll know what this is about. Hilbert learned a great deal from those two in his early days. They talked to each other as equals. By the way, this whole introduction is context. It is supposed to make some sense of the problem list. Charles Matthews 08:50, 30 September 2007 (UTC)

text: Please feel free to work on the article. Charles Matthews 08:50, 30 September 2007 (UTC)
thread_title:  Stilted and pretentious language 

message: "Baseball's Hilbert Problems: 23 Burning Questions".  Heh.  66.127.55.192 (talk) 01:57, 19 February 2010 (UTC)
thread_title:  baseball statistics 

message: The summary section says 11 has an accepted answer, but the table says otherwise  — Preceding unsigned comment added by 98.109.170.24 (talk) 22:20, 9 June 2011 (UTC)
replies: text: Also, the 13th problem in the table says "Unresolved", but is still colored green.  I couldn't figure out how to fix it and I'm not sure I'm not missing something anyway...  — Preceding unsigned comment added by 124.148.188.82 (talk) 10:44, 17 July 2011 (UTC)
replies: text: Ditto problem 22, whose color I changed to pink to match the "Unresolved" in the table. -- Elphion (talk) 01:59, 25 October 2018 (UTC)
thread_title:  Summary 

message: I think it is misleading simply to say, as the article does, that the problem is "too vague" and to leave it at that. Although the problem does have various interpretations, the primary one is surely the one given by Hilbert himself.
:We are asking then, for a geometry in which all the axioms of ordinary Euclidean geometry hold, and in particular all the congruence axioms except the one of the congruence of triangles (or all except the theorem of the equality of the base angles in the isosceles triangle), and in which, besides, the proposition that in every triangle the sum of two sides is greater than the third is assumed as a particular axiom.
In fact, in 1973 Pogorelov gave a complete characterization of such geometries in dimensions 2 and 3. Busemann confirmed that the method can be extended to all dimensions. 
Although the information that it is "too vague" is properly sourced to Gray's summary of the status of the Hilbert problems, my impression on reading  and  is that, in the primary sense intended by Hilbert, as understood by specialists, the problem has been solved to the extent that it can ever be. 
The idea that the problem is too vague is most likely related to the following situation, as described by Busemann:
:The fourth problem concerns the geometries in which the ordinary lines, i.e. lines of an n-dimensional (real) projective space Pn or pieces of them, are the shortest curves or geodesics. Speciﬁcally, Hilbert asks for the construction of these metrics and the study of the individual geometries. It is clear from Hilbert’s comments that he was not aware of the immense number of these metrics, so that the second part of the problem is not at all well posed and has inevitably been replaced by the investigation of special, or special classes of, interesting geometries.
In , Busemann characterizes the problem as "(1) Determine all geometries satisfying these conditions. (2) Study the individual ones." Then he writes: "The discovery of the great variety of solutions showed that part (2) of Problem 4 is not feasible. It is therefore no longer considered as part of the problem. But many interesting special cases have been studied since 1929." Papadopoulos agrees, writing "The second part of the problem is indeed very wide, and because of that it is likely to remain open forever." 
In my view, this should take nothing away from the solution of part (1), and the discovery that there are simply too many solutions to "study the individual ones."
Likewise, the fact that the problem can also be re-interpreted in other ways doesn't alter the fact that the formulation of it that Hilbert stated in the quote above has been solved.
I propose therefore that something be said in the table about Pogorelov's solution. 184.171.221.61 (talk) 08:37, 23 December 2013 (UTC)
replies: text: I suggest to first edit Hilbert's fourth problem to insert the history of the question that you have summarized (Pogorelov's solution included), and then correct the table as "partially resolved" or "resolved" (I have not a clear opinion), with an explaining note similar to that of fifth problem. D.Lazard (talk) 11:28, 23 December 2013 (UTC)
thread_title: Hilbert's fourth problem

message: In the discussion of the 6th problem, no effort is made to distinguish the pre-quantum mechanical, pre-relativistic Physics that Hilbert must have been referring to in 1900 (since neither had been 'discovered' yet) from modern physics. I think the term 'physics' should be changed to 'Classical physics'. (I saw no mention of this in the archived (pre-2007) talk-page section on Problem 6.) It also seems to me, given this qualification, that Problem 6 HAS BEEN COMPLETED. (as stated Kolmogorov's axiomatics seems to complete the problem, but I am NOT an expert in this, so must defer to others.) Thus, the statement made in the article that the the axiomatics of [classical] physics is now more "remote" seems false, although it almost certainly is less important (since classical physics is no longer regarded as a fundamentally adequate basis for describing the physics of our Universe).
I contend:
:1. 'physics' should be qualified as 'classical physics' or perhaps '[classical] physics if in a direct quote. (For the mathematical basis of the Standard Model, see the - yet to be claimed - Clay prize for Yang–Mills existence and mass gap problem) and
:2. "both more remote and " should be deleted (its not only arguably wrong, but 'remote' is a weasle word)173.189.79.42 (talk) 22:17, 15 May 2015 (UTC)
thread_title:  6th Problem - again 

message: The following sentence:
There are two problems that are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem concerns the axiomatization of physics, a goal that twentieth century developments of physics (including its recognition as a discipline independent from mathematics) seem to render both more remote and less important than in Hilbert's time. Also, the 4th problem concerns the foundations of geometry, in a manner that is now generally judged to be too vague to enable a definitive answer.
has no source and I believe it is vague and/or plainly false (as some comments above point out, too). The problems are examined in details, for example, in the book edited by Browder (about 110 pages for Problem 6 !). As another source, the Encyclopedia of Mathematics mentions them, asserting 4 is solved and 6 is very far from solved in any way... (but) there is a great deal of interest and activity in (the axiomatic approach represented by) topological quantum field... (etc.)
In my opinion the sentence should be deleted.
Turning to an argument I know a bit better, it is uncontroversial that the second problem deals with a theory much stronger than Peano Arithmetic (I guess that the definition itself of PA has been given in subsequent years). (Now I see that this has been discussed clearly also in Talk:Hilbert's second problem.) The theory mentioned by Hilbert should deal at least with  real numbers, but at the end Hilbert even mentions "Cantor’s higher classes of numbers and cardinal numbers". It is a quite common opinion that Gödel's incompleteness gives a negative solution to problem 2, though this is not accepted by some. On the other hand, Gentzen deals only with PA; only a much stronger result could be considered as a possible solution to the second problem.  Gentzen-like results, so far, can be considered only as very partial solutions (see, e.g., Simpson, Subsystems of Second Order Arithmetic). As far as I know, a Gentzen-like result for second order arithmetic is presently beyond reach , let apart full set theory! . I suggest to delete the reference to Gentzen as a possible solution. In view of the subsequent formulations of Hilbert program, it is debatable whether Hilbert could have considered Gentzen proof as "finitistic". However, the point is that Gentzen proof deals with a theory which is too weak.
By the way, parts of the section "Ignorabimus" are historically inaccurate: Hilbert program has been developed after the problem list. Paolo Lipparini (talk) 17:40, 11 October 2015 (UTC)
thread_title:  Some issues 

message: Sorting the table of problems by problem number yields a very strange order, with 1st in the middle.Retardednamingpolicy (talk) 13:20, 21 February 2016 (UTC)
replies: text: What do you mean by "1st in the middle"? The numbering and and the resulting sorting of the table are due to Hilbert himself. Most of the problems are referred to in the literature (and often in article titles) by their number. This cannot be changed by Wikipedia nor by anybody. D.Lazard (talk) 14:15, 21 February 2016 (UTC)
replies: text: Huh? I literally mean 1st in the middle. 10-19, then 1, then 20-23, then 2-9. Just try it. Retardednamingpolicy (talk) 14:47, 21 February 2016 (UTC)
replies: text:  On my browser the problems are displayed in the normal ordering. It seems that you try to sort the table with an algorithm that treats the problem numbers as character chains, in which case the ordering is correct, as usually the alphabetic characters appear after the digits in the lexical ordering. Wikipedia tables are not designed for being sorted by external programs nor for being sorted in a different way. D.Lazard (talk) 15:38, 21 February 2016 (UTC)
replies: text: Did you actually try sorting by problem number, as in click on Problem in the header? For me the table is displayed in the proper ordering as well when the page first loads, but if you wanna sort by something else and then revert to the original sorting, that's not possible. I tried on Chrome, Firefox and Internet Explorer, Windows 8.1, so clearly it's not just me.Retardednamingpolicy (talk) 16:58, 21 February 2016 (UTC)
replies: text:  Understood. The table was mistakenly marked as "sortable". I have edited it for fixing the problem. D.Lazard (talk) 17:19, 21 February 2016 (UTC)
thread_title:  Table sorting glitch 

message: Is Hilbert's 16th problem really too vague to be resolved? The problem seems well-posed to me, and the article on the 16th problem also seems to treat it (and attempts to solve it) seriously. Same goes for the various journal articles dealing with the problem. I'm tagging the statement in the summary section as dubious, and reverting the status back to "unresolved" for now. Banedon (talk) 02:54, 8 August 2016 (UTC)
thread_title:  Dubious 

message: The books contain information of some generality. For example, the 10th problem is directly connected to (interpretations) of the 2nd problem, e.g. "it concerned a decision problem for Diophantine equations that was related to unpublished results he [Goedel] had obtained years earlier as corollaries to his incompleteness theorems." (e.g. from Dawson "Logical Dilemmas" 1997:238). But if they are not there already these references probably could be moved to the relevant sub-articles (the Matijasevich reference is indeed in the 10th problem article). I'd prefer them to stay, but I'll leave your edit in place and won't dispute the matter further. Bill Wvbailey (talk) 17:38, 10 August 2016 (UTC)
replies: text: I'm going to put them back.  This article is a bit of an anomaly in that it gathers together quite disparate topics in a way that might ordinarily violate the principle that an article should be about a single topic, except for the very noteworthy connection of going through Hilbert's challenge.  When a reader drills down into the article, he/she may very well want references on one of the individual problems, even if that reference is not talking about the subject of the article as a whole, meaning the problems considered together. --Trovatore (talk) 20:36, 10 August 2016 (UTC)
replies: text: The problem with that is that there are 23 problems, and if we allow references to individual ones in this general article, then the number of possible references balloons quickly. We also have articles on each individual problem, which are the obvious places to look if one is interested in those. The current article even says "for details on the solutions and references, see the detailed articles that are linked to in the first column" just before the table. I think leaving the references out is preferable. Banedon (talk) 00:52, 11 August 2016 (UTC)
replies: text: The references section is of quite manageable size; I don't think we have a problem with too many. --Trovatore (talk) 01:27, 11 August 2016 (UTC)
replies: text: That's right now, but what about in the future? If we keep these references then adding references to problem 1, problem 2, etc would make sense and "improve" the article. The Riemann Hypothesis article, Hibert's problem 8, contains many such references. Is there a reason not to copy those references over? Banedon (talk) 06:04, 11 August 2016 (UTC)
replies: text: Well, we could certainly copy one or two especially useful ones; I can't see any problem with that.  Sometimes WP has a tendency to overdo references sections and they get hard to find anything in, but I don't think we're near that line in this particular article. --Trovatore (talk) 06:36, 11 August 2016 (UTC)
thread_title: Removed references 

message: According to Hilbert's eighteenth problem, the 18th problem has three parts, the first of which was answered by Bieberbach. This page only lists two parts. Is there a reason for that? Should this page be updated? 178.11.76.129 (talk) 09:14, 31 July 2020 (UTC)
thread_title: 18th problem: 3 parts?

message: Reference 7 (http://www.networkworld.com/community/node/33361) is a dead link that just goes to a "Page not Found" page.    I don't know what, if anything, it should be replaced by.  — Preceding unsigned comment added by 82.40.177.40 (talk) 14:53, 8 February 2021 (UTC)
thread_title:  Reference 7 404s. 

message: It seems this article struggles somewhat to talk about all the problems together in a properly explanatory manner. Hilbert was a massively influential mathematician and all of his 23 problems invoke incredibly complex topics. The relevance of these topics cannot meaningfully be explained by simply throwing in a links to Axiomatic set theory, for example. Multiple of the problems are both noteworthy enough and difficult enough to explain that they deserve their own articles. EditorPerson53 (talk) 12:19, 29 April 2023 (UTC)
replies: text: Each of the Hilbert's problem has already its own article. They can be reach by clicking on the first column of the table. : everybody agree with this assertion, and it is probably for this reason that axiomatic set theory is not linked to in this article. D.Lazard (talk) 12:36, 29 April 2023 (UTC)
thread_title:  Create articles for each problem?